12 Applications: Games of Chance; PokerIn a 5 card hand from a deck of 52, there are (52*51*50*49*48)/(5*4*3*2*1) different possible hands. (Order doesn’t matter). 2,598,960 possible hands.How many of these hands have 4 aces? 48 = the 4 aces plus any of the remaining 48 cards.

13 Some Poker HandsFull House – 3 of one kind, 2 of another. (Also called a “boat.”)Royal Flush – Top 5 cards in a suitFlush – 5 cards in a suit, not sequentialStraight Flush – 5 sequential cards in the same suit suitStraight – 5 cards in a numerical row, not the same suit4 of a kind – plus any other card

15 The Dead Man’s HandThe dead man’s hand is 5 cards, 2 aces, 2 8’s and some other 5th card (Wild Bill Hickok was holding this hand when he was shot in the back and killed in 1876.) The number of hands with two aces and two 8’s is = 1,584The rest of the story claims that Hickok held all black cards (the bullets). The probability for this hand falls to only 44/ (The four cards in the picture and one of the remaining 44.)Some claims have been made about the 5th card, but noone is sure – there is no record.

20 More InformationDeduce: Since P(+|D)=.98, we know P(–|D)= because P(-|D)+P(+|D)=1 [P(–|D) is the P(False negative).Deduce: Since P(–|N)=.95, we know P(+|N)= because P(-|N)+P(+|N)=1[P(+|N) is the P(False positive).Deduce: Since P(D)=.005, P(N)=.995 because P(D)+P(N)=1.

24 Random VariableDefinition: Maps elements of the sample space to a single variable:Assigns a number to   Discrete: Payoff to poker handsContinuous: Lightbulb lifetimesMixed: Ticket sales with capacity constraints. (Censoring)

40 Model for Light Bulb LifetimesThis is the exponential model for lifetimes. The model is f(time) = (1/μ) e-time/μ

41 Model for Light Bulb LifetimesThe area under the entire curve is 1.0.

42 Continuous DistributionThe probability associated with an interval such as 1000 < LIFETIME < equals the area under the curve from the lower limit to the upper.A partial area will be between 0.0 and 1.0, and will produce a probability.

43 Probability of a Single Value Is ZeroThe probability associated with a single point, such as LIFETIME=2000, equals 0.0.

46 Common Continuous RVsContinuous random variables are all models; they do not occur in nature. The model builder’s toolkit:Continuous uniformExponentialNormalLognormalGammaBetaDefined for specific types of outcomes

58 Censoring and TruncationObservation mechanism. Values above or below a certain value are assigned the boundary valueApplications, ticket market: demand vs. sales given capacity constraints; top coded income dataTruncationObservation mechanism. The relevant distribution only applies in a restricted range of the random variableApplication: On site survey for recreation visits. Truncated PoissonIncidental truncation: Income is observed only for those whose wealth (not income) exceeds $100,000.

67 Textbooks Provide Tables of Areas for the Standard NormalEconometric Analysis, WHG, 2008, Appendix G, page 1093, Rice Table 2Note that values are only given for z ranging from 0.00 to No values are given for negative z.

70 Standard Normal Distribution FactsThe random variable z runs from -∞ to +∞(z) > 0 for all z, but for |z| > 4, it is essentially 0.The total area under the curve equals 1.0.The curve is symmetric around 0. (The normal distribution generally is symmetric around μ.)

71 Only Half the Table Is NeededThe area to left of 0.0 is exactly 0.5.

72 Only Half the Table Is NeededThe area left of 1.60 is exactly 0.5 plus the area between 0.0 and 1.60.

73 Areas Left of Negative ZArea left of -1.6 equals area right of +1.6.Area right of +1.6 equals 1 – area to the left of